In the last chapter – the introduction to part 2 of this book – we set the stage for a new, more powerful foundation for our language of reasoning. We’re officially giving up on machine diagrams, because although they can help understanding of simple concepts, they lack the precision necessary to describe things at the level of abstraction we are just beginning to uncover.

We will call our new “version” of machine diagrams symbolic computation. It’s an apt name, since what it provides us is a mechanism for describing the computation of symbols. We don’t have anything more than a loose intuition behind what it means to “compute” something, but we’ll ignore that for a while. As a good bet, anything the universe is capable of doing is probably computable (if it weren’t, how does the universe figure out what the answer is?).

There are reasonable objections to this claim, but we will try not to spill ink on defending or refuting them. All I’m asking is to keep an open mind; it seemed unlikely beforehand that we could have built a device capable of remembering information when we started in chapter 1 with machines and wires, but as we built more and more tools, eventually we managed to get there. I’m not saying that everything the universe does is easy to compute, merely that it does indeed “compute” it somehow – for some definition of “compute”, at least. I offer no promises that we’ll be able to compute morality, for example, but I promise it will seem less daunting by the time we’re finished.

To make good on a previous promise of mine, in this chapter we’ll look at how we can migrate the notion of “wires” from our machine diagram foundations to our new symbolic computation framework.

Before we do that, however, the first question we should ask is “what purpose did wires serve before?” Our original answer to that question was that they served no purpose other than to connect one machine to another. This is true, but we also subtly in another way – we conflated the connections between machines with the values that they held. This wasn’t intentional, but it was just that our machine diagrams lacked the expressiveness to describe “values at rest”.

Moving forwards, we’ll differentiate between these two notions. This means that we can discuss information at rest (values of a type), as well as means of connecting machines (machine composition). Because we’ve already discussed types and values, we move on to look at machine composition.

Symbolic Computation

Symbolic computation is named thus because it defines computation in terms of opaque symbols. This means that our values, in their most basic incarnation, have no internal structure. Values are nothing more than labels that we humans put on distinct things. The only thing we know about these distinct things is that they are different from one another, and that we can differentiate them at will. We know that On is different than Off, but the labels we have chosen for them are nothing more than convention. We could have called them Flubbix and Rathcold, and the theory would be none-the-wiser.

A point we should make explicit here is that because values are distinct, there must never be an ambiguity in determining which symbol we are discussing. That means while zee might be another label for On, it cannot be a label for both OnandOff. This was the rule we were implicitly enforcing when we said of machine diagrams that their “function tables never lie”. To reiterate, that means that every machine’s output must be defined for every possible input it might receive. We say of such a machine that it is total.

Takeaway: A total function is one that has output defined for each and every possible input it might accept.

But enough philosophizing. I promised to talk about how to compose machines together. As a beginning exercise, let’s define one machine in terms of another. We want to construct this machine in our new system:

Let’s say we have the symbolic definition of Blah:

blah : Bool -> n -> Bool
blah On _ = On
Blah Off _ = Off

which corresponds with this function table:

Input A

Input B

Output

0

_

0

1

_

1

blah is obviously just a machine that ignores its second wire. We use an underscore _ to indicate that we don’t care which value that wire holds.

But notice here that we’re doing more work than is strictly necessary. We could instead have written a semantic symbol table:

Output

A

which says that the output is always just the input A. We can rewrite blah in this manner, too:

blah : Bool -> n -> Bool
blah a _ = a

Here, we’ve used the symbola to refer to the value coming in on the A input, regardless of what it might be. We call this a a binding, because it binds to the value coming in on the input wire. If that value is On, then a would currently act as a synonym for the value On. There’s nothing new here, it’s just a new framing to wrap your head around.

Getting back to our diagram:

As this diagram shows, Boring is nothing more than a Blah hidden inside of a box. Boring is thus a (very boring) composition of Blah. If we had Blah’s function table, we could arduously and painstakingly copy it to define Boring’s, but this breaks our desired abstraction semantics. In our machine diagrams, we didn’t need to know what Blah’s function table was, and that was kind of the whole point.

We can instead describe boring as a symbolic computation like this:

boring : Bool -> n -> Bool
boring a b = blah a b

This definition should be read like this:

boring is a machine which takes two inputs: one a Bool and the other some polymorphic n. This machine then outputs a Bool.

It is defined as binding a to the first input (the Bool), and b to the second (the polymorphic n), and passing them as the first and second inputs to the blah machine, respectively.

Quite a mouthful, isn’t it? That’s why we use the more terse symbolic computation definition. It saves a lot of time, and after you get good at reading it, you’ll appreciate the level of precision it affords us.

Looking at the definition boring a b = blah a b is quite telling, in its own right. If you read that in your most exciting “math” voice, it says that boring is simply equal to blah. Which is exactly what our machine diagram said, too! Kinda neat, isn’t it?

Let’s try a more sophisticated example. Recall our definition of a nand gate:

We can express this symbolically:

nand : Bool -> Bool -> Bool
nand a b = not (and a b)

This reads a little backwards from what you might expect, but it makes some sense if you think about it. Remember that nand a b = means we are binding a and b to the inputs of nand, so it seems fair enough that we should put the input to notafter it. We use parenthesis to indicate that it is the entire and a b (the output of the and gate, with a and b as inputs) that we would like to give as an input to our not gate. Make sense?

Let’s do another.

would be described like this:

nandn a b = not (and (not a) b)

Reading these things is a bit of an art, but the trick is to always start from the innermost set of parenthesis and work your way outwards. Here, we’re passing the not of a as the first input to our and, with b being the second input. Then we’re taking the output of the entire and gate, and giving it as an input to not.

In the next chapter, we’ll discuss the evaluation semantics of this symbolic computation – which is to say that we’ll see how we can actually perform “computation” with this method.

Exercises

Give two symbolic definitions for xor – one in terms of its function table, and one as a composition of not, or and and gates (recall the composition from chapter 6).

Give a compositional symbolic definition for the Cout machine (recall the implementation from chapter 6).